Characterization of a hyperperfect group

Abstract

In this work, we have introduced the notion of hyperperfect group. A group of order n is said to be hyperperfect if there exists a natural number k such that n-1 = k[σ(n)-n-1] where σ(n) denotes the sum of positive divisors of n. We have also established a condition under which a cyclic group is hyperperfect. We have established that no group of prime order is hyperperfect and investigated the same for groups of various non-prime order. We have also determined an upper bound of the order of a hyperperfect group.